Received: 17 July 2017 – Discussion started: 31 July 2017Revised: 16 December 2017 – Accepted: 31 December 2017 – Published: 6 February 2018
Abstract. Bedrock channel slope and width are important parameters for setting bedload transport capacity andfor stream-profile inversion to obtain tectonics information. Channel width and slope development are closelyrelated to the problem of bedrock channel sinuosity. It is therefore likely that observations on bedrock channelmeandering yields insights into the development of channel width and slope. Active meandering occurs whenthe bedrock channel walls are eroded, which also drives channel widening. Further, for a given drop in elevation,the more sinuous a channel is, the lower is its channel bed slope in comparison to a straight channel. It can thusbe expected that studies of bedrock channel meandering give insights into width and slope adjustment and viceversa. The mechanisms by which bedrock channels actively meander have been debated since the beginning ofmodern geomorphic research in the 19th century, but a final consensus has not been reached. It has long beenargued that whether a bedrock channel meanders actively or not is determined by the availability of sedimentrelative to transport capacity, a notion that has also been demonstrated in laboratory experiments. Here, this ideais taken up by postulating that the rate of change of both width and sinuosity over time is dependent on bedcover only. Based on the physics of erosion by bedload impacts, a scaling argument is developed to link bedrockchannel width, slope and sinuosity to sediment supply, discharge and erodibility. This simple model built onsediment-flux-driven bedrock erosion concepts yields the observed scaling relationships of channel width andslope with discharge and erosion rate. Further, it explains why sinuosity evolves to a steady-state value andpredicts the observed relations between sinuosity, erodibility and storm frequency, as has been observed formeandering bedrock rivers on Pacific Arc islands.
1 Introduction
Bedrock channels are the conveyer belts of mountain re-gions. Once sediment produced on hillslopes by mass wast-ing reaches a channel, it is evacuated along the stream net-work. In the process, the moving particles act as tools forbedrock erosion and the river adjusts until it reaches a steadystate. Then, channel morphology, parameterized for exam-ple by the width and bed slope of the channel, stays con-stant over time, and the vertical erosion rate adjusts to matchtectonic uplift. In turn, width and slope determine the trans-port capacity of the channel and thus the channel’s efficiencyin evacuating sediment. As a result, channel long profilesand width can be used as indicators for local tectonic pro-
cess rate, and uplift rates and histories can in principle becalculated from morphologic characteristics of the channelnetwork (e.g. Kirby and Whipple, 2001; Roberts and White,2010; Wobus et al., 2006b). Conventionally, the inversion ofchannel morphology to obtain tectonic information has fo-cussed on long profiles or slope, despite the observation thatchannel width also adjusts to tectonic forcing (e.g. Duvall etal., 2004; Lavé and Avouac, 2001; Yanites et al., 2010). Forreliable inversion we thus need a model that can predict theeffects of uplift both on channel width and slope (e.g. Tur-owski et al., 2009) and possibly other morphologic parame-ters.
Published by Copernicus Publications on behalf of the European Geosciences Union.
30 J. M. Turowski: Alluvial cover controlling of bedrock channels
Davis (1893) sparked a long-standing debate in geomor-phology when he described the meanders of the Osage Riveras inherited from a prior alluvial state of the channel. Al-though this explanation is still frequently encountered to ex-plain why bedrock channels are sinuous, even Davis’ con-temporaries argued that active meandering occurs in incisedchannels (e.g. Winslow, 1893). By now, numerous field ob-servations of features such as cut-off meander loops and gen-tle slip-off slopes in inner meander bends have confirmedthat actively meandering bedrock channels exist and are com-mon (e.g. Barbour, 2008; Ikeda et al., 1981; Mahard, 1942;Moore, 1926; Seminara, 2006; Tinkler, 1971). However, themechanics of bedrock river meandering are still debated andhave recently attracted research interest (e.g. Johnson andFinnegan, 2015; Limaye and Lamb, 2014), since the mean-dering problem is closely related to the problems of terraceformation, lateral planation, gorge eradication and bedrockchannel width (cf. Cook et al., 2014; Finnegan and Balco,2013; Turowski et al., 2008a). In fact, active meandering isdependent on the lateral erosion of the channel walls and istherefore directly related to the adjustment of channel width.Similarly, meandering lengthens the channel over a givendrop of height and thereby reduces bed slope. Thus, it seemslikely that observations on bedrock channel sinuosity are in-formative also for the study of channel width and slope andvice versa. While the power-law scaling of channel widthand slope with discharge with typical exponents of ∼ 1/2,positive for width and negative for slope, is widely acknowl-edged (e.g. Lague, 2014; Snyder et al., 2003; Whipple, 2004;Whitbread et al., 2015; Wohl and David, 2008), observationsof the scaling relationships of sinuosity are less commonlydiscussed. In a detailed study of Japan, Stark et al. (2010)demonstrated that lithology poses a first-order control on thesinuosity of actively incising bedrock channels, with weaksedimentary rocks displaying higher values of a regionalmeasure of sinuosity than volcanic or crystalline lithologies.Once this influence was accounted for, a positive trend ofsinuosity with the variability in precipitation emerged, quan-tified by typhoon-strike frequency or by the fraction of dayswith rainfall exceeding a threshold. This positive trend withstorm frequency could generally be confirmed for other is-lands of the Pacific Arc, including Taiwan, Borneo, NewGuinea and the Philippines (Stark et al., 2010). The predic-tion of the relationships observed by Stark et al. (2010) re-mains a benchmark for any theory of bedrock channel me-andering, but an explanation is lacking so far. Further, in ad-dition to observations on channel bed slope and width, thesinuosity scaling provides another line of evidence for thevalidation of general models of bedrock channel morphol-ogy.
Sinuosity increases when, within a channel bend, the bankat the outer bend erodes faster than at the inner bend. Allu-vial meander theory relates this imbalance in lateral erosionto hydraulics (e.g. Edwards and Smith, 2002; Einstein, 1926;Ikeda et al., 1981). Within the bend, there are higher flow
speeds in the outer bend than in the inner bend. Erosion rateand therefore the meander migration rate is assumed to bedependent on the velocity difference. In contrast, in manybedrock channels, erosion is driven by particle impacts inthe two most common fluvial erosion processes: pluckingand impact erosion (e.g. Beer et al., 2017; Chatanantavet andParker, 2009; Cook et al., 2013; Sklar and Dietrich, 2004).Abrasion means the erosion due to impacts of moving bed-load particles. Plucking means the removal of larger blocksof rock by hydraulic forces. In the latter process, impactsdrive crack propagation and thus the production of pluckableblocks, which is also known as macro-abrasion (Chatanan-tavet and Parker, 2009). In environments where particle im-pacts drive erosion, the outer bends of meanders are partic-ularly prone to erosion as particle trajectories detach fromflow lines and can thus impact the walls (e.g. Cook et al.,2014). If bedrock channel sinuosity is indicative of past cli-mate, as Stark et al. (2010) suggested, then bedrock chan-nels need the ability to first adjust to the required sinuosityand second keep this sinuosity constant over long time peri-ods, while continuing vertical incision. The latter feat can beachieved either by stopping lateral erosion once the requiredsinuosity is reached or by maintaining a balance of thoseprocesses that increase sinuosity and those that decrease it.The only known mechanism for decreasing sinuosity is me-ander cut-off. However, cut-off can only occur if the chan-nel meanders actively, and it is only effective when sinuos-ity is high. The sinuosity of bedrock channels observed byStark et al. (2010) span a wide range of values, including lowsinuosities that cannot be kept steady with recurring cut-off.Thus, it seems unlikely that the cut-off mechanism can bal-ance lateral erosion rates at low sinuosity to achieve a steadystate. The argument suggests that channels cease or at leaststrongly decrease active meandering once they have reachedthe steady-state sinuosity, but why they do this is an unsolvedproblem. This raises the question as to when and why somebedrock channels actively meander while others do not. Ingeneral, two lines of argument have been proposed to answerthis question.
The first line of argument asserts that the process ofbedrock erosion controls lateral erosion rates, and locallithology determines this process and thus whether a chan-nel actively meanders or not. Johnson and Finnegan (2015)compared two bedrock channels in the Santa Cruz Moun-tains, California, USA: one actively meandering in a mud-stone sequence, the other one incising without meanders intoa sandstone. While both lithologies showed similar strengthwhen dry, the mudstone lost strength through slaking dueto wetting–drying cycles and could thereafter be eroded byclear water flows. In this case, essentially, active meanderingcould be achieved by a similar hydraulic mechanism as hasbeen described for alluvial streams (e.g. Edwards and Smith,2002; Ikeda et al., 1981; Seminara, 2006). Moore (1926)likewise described an influence of lithology on the mean-ders of streams on the Colorado Plateau – there, meanders
J. M. Turowski: Alluvial cover controlling of bedrock channels 31
can be found in sandstone units, while in weaker shales, thevalleys are wide and straight. However, Moore (1926) didnot describe different erosion mechanisms (e.g. slaking, im-pact erosion) for the two lithologies, and it is unclear whatcauses the different channel behaviour in his study region.While the slaking mechanism should be more efficient in avariable climate due to more frequent wetting–drying cycles,in line with Stark et al.’s (2010) observations, it fails to ex-plain why a stream can continue incising while maintaininga constant sinuosity. Further, Stark et al. (2010) describedsinuous bedrock channels in a range of lithologies, includ-ing hard crystalline rock, where slaking erosion is likely notimportant.
The second line of argument builds on the relative avail-ability of sediment in the channel. In resistant bedrock, ero-sion is driven by the impacts of moving particles in the twomost common fluvial bedrock erosion processes, abrasionand plucking. The increasing erosion rate with increasingrelative sediment supply is known as the tools effect (e.g.Cook et al., 2013; Sklar and Dietrich, 2004). Conversely, sta-tionary sediment residing on the bed can protect the bedrockfrom impacts. This is known as the cover effect (e.g. Sklarand Dietrich, 2004; Turowski et al., 2007), which has beenargued to play a key role in the partitioning of vertical to lat-eral erosion (e.g. Hancock and Anderson, 2002; Turowski etal., 2008a). Moore (1926) suggested that whether a bedrockriver actively meanders or not depends on the relative avail-ability of sediment, a notion that was later investigated exper-imentally by Shepherd (1972). In Shepherd’s (1972) experi-ments, a sinuous channel was cut into artificial bedrock madeof sand, kaolinite and silt, which was not erodible by clearwater flow. Base level, water discharge and sediment supplywere kept constant over the entire run time of 73 h. At first,all sediment could be entrained by the flow and the chan-nel cut downwards, without changing the planform pattern.But as the channel bed slope declined due to erosion over thecourse of the experiment, patches of sediment formed on theinside bends and the channel started to widen and to mean-der actively. Shepherd (1972) suggested that lateral erosionrates stayed similar throughout the entire run, while verti-cal erosion rates declined due to the increasing importanceof the cover effect. Thus, at first, lateral and vertical ero-sion were balanced such that channel width remained con-stant over time, while the later decrease in vertical incisionled to channel widening and ultimately migration and activemeandering.
Shepherd’s (1972) experimental observations point to thefundamental importance of bed cover in setting bedrockchannel width and meandering dynamics. In this paper, I de-velop a physics-based scaling argument to explain the ob-served scaling of bedrock channel width, slope, and sinuos-ity. The argument is motivated by the behaviour of the exper-imental channel of Shepherd (1972) and is built on the funda-mental assumption that bed cover controls lateral erosion. Itexploits general considerations and observations about bed-
load transport and process knowledge of fluvial bedrock ero-sion. Since channel morphology is set by the partitioning oferosion between bed and banks, the problem is approachedby assessing under which conditions lateral erosion can oc-cur and how these conditions relate to channel bed cover. Thephysical considerations lead to a model of incising channelswith stable width, slope and sinuosity. Model predictions arecompared to observed scaling relationships of bedrock chan-nel width and slope with discharge, drainage area and ero-sion rate and to the sinuosity scaling observed by Stark etal. (2010).
2 Model development
Previous attempts at predicting bedrock channel morphologycan be grouped into four classes (Shobe et al., 2017, alsogave a recent overview):
i. 1-D models using a shear stress or stream power for-mulation (e.g. Seidl et al., 1994; Whipple, 2004). Thesemodels capture the fundamental scaling of slope withdischarge and, to an extent, of slope with erosion rate,but need to make assumptions on width–discharge scal-ing for closure (see Lague, 2014, for a review). Zhanget al. (2015) described a morpho-dynamic model thatalso captures alluvial dynamics and includes both toolsand cover effects. However, this model is restricted tochannels with macro-rough beds, i.e. topography witha relief that is a substantially larger than the dominantgrain size.
ii. 1-D models that treat channel width explicitly, but, in-stead of assuming a width–discharge scaling, make analternative assumption to close the system of equations.Suggested assumption have been a constant width-to-depth ratio (Finnegan et al., 2005) or the optimizationof energy expenditure (Turowski et al., 2007). Thesemodels have been proposed assuming a shear stress orstream power erosion law (Finnegan et al., 2005; Tur-owski et al., 2009), as well as sediment-flux-dependenterosion laws including either just the cover effect (Yan-ites and Tucker, 2010) or both tools and cover effects(Turowski et al., 2007). For the shear stress erosionmodel, the closing assumption has at least been partiallyvalidated against models treating the cross-sectionalevolution of a channel (Turowski et al., 2009). Althoughthese models can predict a range of observed scaling re-lations, especially if sediment flux effects are includedin the erosion model (see Turowski et al., 2007; Yanitesand Tucker, 2010), they suffer from a lack of physics-based arguments for connecting lateral erosion to chan-nel morphology and from the essential arbitrariness ofthe closing assumption.
iii. 2-D models that explicitly model some aspects ofthe width dynamics. For a shear stress erosion law,
32 J. M. Turowski: Alluvial cover controlling of bedrock channels
Stark (2006) used a slanted trapezoidal channel shape,while Wobus et al. (2006a) and Turowski et al. (2009)described models with a fully adjustable channel crosssection. Lague (2010) used a trapezoidal cross sectionand included the cover effect in his formulation. Thesuccess of these models in predicting scaling relation-ships is similar to that of the models of class (ii), butnone of the models published so far include all aspectsof the current understanding of the process physics offluvial bedrock erosion. Further, none of these mod-els properly treat fully alluviated beds, where alluvialchannel processes dominate, which can strongly affectlong-term erosional dynamics and channel adjustmenttimescales (cf. Turowski et al., 2013).
iv. 3-D models that, to some extent, resolve the interac-tion of hydraulics and sediment transport and their ef-fect on bedrock erosion (e.g. Inoue et al., 2016; Nelsonand Seminara, 2011, 2012). These models are generallynumerically expensive and have not been used to inves-tigate scaling relations on the reach to catchment scale.
As a result, we lack a model that is rooted in the currentunderstanding of process physics and can predict channelwidth, slope and sinuosity on the catchment scale. Here, in-spired by the experiments described by Shepherd (1972), Iput forward the fundamental postulate that the partitioningbetween lateral and vertical erosion, and therefore width ad-justment and sinuosity development, is controlled by a sin-gle variable: bed cover. Parameters such as sediment supply,river sediment transport capacity and bed topography directlycontrol cover, but they only indirectly control the distribu-tion of erosion by altering bed cover. Formalizing the ob-servations made in Shepherd’s (1972) experiments, we canmake the following statements: (i) at low degrees of cover,channel width stays constant and the channel does not mean-der actively, and (ii) channel widening and active meanderingcommences when a threshold cover is exceeded. In Sect. 2.1,based on considerations based on the physics of erosion byparticle impacts and of bedload transport, I develop a scalingargument for bedrock channel width. In Sect. 2.2, the slopeof the channel is discussed. In Sect. 2.3, the argument is ap-plied to the development of bedrock channel sinuosity.
2.1 Lateral erosion and bedrock channel width
Consider a straight bedrock channel with sub-vertical walls.The general direction of water and particle discharge is par-allel to the walls, although we can expect some lateral mo-tion due to secondary currents and turbulent fluctuations. Asbedrock erosion is achieved by particle impacts, the require-ment for lateral erosion is a sideward deflection of travellingparticles such that they (i) reach and impact the wall and,(ii) upon impact, have enough energy to cause damage to therock. Lateral motion of sediment particles can be driven by
secondary currents, turbulent fluctuation and momentum dif-fusion (e.g. Diplas et al., 2008; Parker, 1978), cross-streamdiffusion of particle paths (Seizilles et al., 2014), gravita-tionally driven migration on cross-sloping beds (e.g. Parkeret al., 2003), or sideward deflection by obstacles on the bed(Beer et al., 2017; Fuller et al., 2016). For given conditions– hydraulics, bed morphology, sediment supply and graincharacteristics – we can define a sideward deflection lengthscale dxy for every point on the bed, which depicts the maxi-mum distance a particle can be deflected sideward while stillcausing bedrock wall erosion. This length scale should bea function of hydraulics or transport capacity, channel bedslope, channel curvature, bed roughness, sediment proper-ties (size, shape, density), and possibly of the erodibility ofthe bedrock via the threshold for erosion. Crucially, it canbe expected that dxy can vary considerably over short dis-tances both along and across the channel, depending on bedtopography and the local distribution of roughness and al-luvial cover. For the construction of a reach-scale model ofbedrock channel morphology, we need to first find the rele-vant point within each cross section and the correspondingdxy that determines lateral erosion (which we call dx) andthen the relevant cross section and the corresponding dx thatdetermines channel width in the reach (which we call d). Fora given channel, the propensity to lateral erosion then de-pends on the ratio of the sideward deflection length scale dand the channel width W (Fig. 1). In a channel with a widthmuch larger than d , only bedload moving close to the walls– or more precisely, within a distance d of the walls – cancontribute to lateral erosion. In contrast, in a channel withW ∼ d, all bedload can contribute to lateral erosion.
In general, a bedrock channel widens only when bedloadparticles impact the walls, i.e. in the framework proposedabove that some bedload is moving within a distance d fromthe walls. For the purpose of illustration, consider a narrow,straight bedrock channel with W ∼ d (Fig. 1). Due to fre-quent particle impacts on the walls, lateral erosion rates arehigh and the channel widens. This leads to a decrease in theareal sediment concentration and thus a decrease in the num-ber of bedload particles that can cause lateral erosion. Atsome point bedload impacts on the wall become so unlikelythat widening ceases. The channel has reached a steady-statewidth. However, this argument does not capture the entirestory, since we have neglected vertical incision. Next, this as-pect will be included in the consideration and the ratio d/Wwill be tied to one of the common observables in bedrockchannel morphology: the covered fraction of the bed C (e.g.Sklar and Dietrich, 2004; Turowski and Hodge, 2017).
The relative efficiency of lateral to vertical erosion hasbeen tied to bed cover in conceptual–theoretical arguments(e.g. Hancock and Anderson, 2002; Moore, 1926), experi-mental observations (e.g. Finnegan et al., 2007; Johnson andWhipple, 2010; Shepherd, 1972) and field studies (e.g. Beeret al., 2016; Johnson et al., 2010; Turowski et al., 2008a).Using a combination of experiments and modelling, it has
J. M. Turowski: Alluvial cover controlling of bedrock channels 33
Figure 1. Illustration of how the sideward deflection length scaled and the channel width interact to determine lateral erosion. Thedashed vertical line shows the relevant deflection point within thecross section. Top: in a narrow channel, particles that are laterallydeflected a distance d may hit the wall and cause erosion. The chan-nel widens. Centre: in a wide channel, the deflected particles donot reach the wall. No lateral erosion occurs. Conversely, few par-ticles travel over the bedrock bed near the wall. Sufficient tools todrive the vertical erosion of the bed are only available within thedistance d of the deflection point. An inner channel with the steady-state width is formed. Bottom: in a steady-state channel, the channelwalls are just out of reach of the deflected particles.
been argued that the fraction of covered bed area is an ade-quate proxy for the reduction in erosion due to the shieldingeffect of sediment on the reach scale (Turowski and Bloem,2016). Consequently, cover C is commonly defined as thecovered bed area fraction, i.e. the bed area covered by sed-iment Acover divided by the total bed area of the consideredreachAtot. Normalizing by the length of the considered reachL, we can write C also as a ratio between two length scales,the relevant covered width Wcover (which could be a reachaverage or the covered width for the cross section relevantfor setting lateral erosion rates) and the channel width W .
C =Acover
Atot=Acover/L
Atot/L=Wcover
W(1)
At low sediment supply, cover is low to non-existent. Suf-ficient tools for incision are available only where the particlestream concentrates. There, an inner channel is formed, andso the channel narrows (e.g. Finnegan et al., 2007; Johnsonand Whipple, 2010). To a similar effect, in wide channels,several longitudinal grooves tend to form at low sedimentsupply (Inoue et al., 2016; Wohl and Ikeda, 1997). One ofthese draws most sediment and water and, after some time,develops into an inner channel that captures the entire wa-ter and sediment supply. At high sediment supply, the bed iscovered by sediment, which reduces vertical erosion to zero.Lateral erosion occurs in a strip just above the cover, wherebedrock is exposed and tools are abundant (Beer et al., 2016;Turowski et al., 2008a). The channel widens. We can formal-ize the observations outlined above by relating the rate ofchange of channel width, dW/dt , to relative sediment sup-
Figure 2. Schematic relation between the rate of change of widthdW/dt (black line) and sinuosity dσ/dt (dashed line) with relativesediment supplyQsQt. At low supply, no sediment particles impactthe walls; the channel narrows and does not meander actively. Athigh supply, frequent sediment impacts on the channel walls drivelateral erosion, leading to channel widening and active meandering.At the critical cover, the rate of change of width is zero. The exactposition of this point depends on absolute channel width.
ply Q∗s , which is the ratio of bedload supply Qs to transportcapacity Qt (Fig. 2). At Q∗s = 0, lateral erosion and there-fore dW/dt is also zero, due to the lack of erosive tools. Forsmall Q∗s , the channel narrows and dW/dt must be negative.For high Q∗s , the channel widens and dW/dt must be posi-tive. Since cover C is generally related to Q∗s (e.g. Sklar andDietrich, 2004; Turowski and Hodge, 2017; Turowski et al.,2007), a similar relationship must arise between dW/dt andcover. At a critical value, Q∗c or Cc, the channel behaviourswitches from narrowing to widening and dW/dt = 0. Thisis the only point where the channel both has a steady widthand incises vertically with a finite erosion rate. At the criticalcover, the distance of bedload particles from the walls needsto be equal to the sideward deflection length scale d. If d islarger than this typical distance, frequent impacts will occuron the channel walls and the channel widens (Fig. 1). If itis smaller, few bedload particles move in the vicinity of thewalls, leading to a lack of erosive tools, and the bed near thewalls is not eroded. An inner channel forms for which theabove condition is true.
As can be seen from the following argument, the criticalcover Cc must depend on channel width and should indeedscale with d/W . Chatanantavet and Parker (2008) demon-strated with experiments that in wide straight channels inthe cover-dominated domain, alternating gravel bars formed.Inoue et al. (2016) modelled this situation and found thata meandering thread of alluvial material between alternat-ing submerged gravel bars migrates downstream over uni-formly eroding bedrock, leading to a channel with a symmet-ric bedrock cross section. From studies on alluvial rivers it isknown that the main path of bedload particles in a straightchannel with submerged bars is offset from the main path
34 J. M. Turowski: Alluvial cover controlling of bedrock channels
of water and the thalweg (e.g. Bunte et al., 2006; Dietrichand Smith, 1984; Julien and Anthony, 2002). Gravel bedloadmoves across the bar, enters the thalweg at the bar centre,traverses it and climbs the next downstream bar at its head(Fig. 3). Similarly, it has been observed that in a partiallyalluviated bedrock channel, sediment moves from patch topatch or from bar to bar (Ferguson et al., 2017; Hodge etal., 2011). However, the precise bedload path over partiallycovered bedrock has not yet been described. For the follow-ing argument, I make two main assumptions. (i) The bedloadpath determined by Bunte et al. (2006) for gravel bed chan-nels with alternating submerged bars applies also to bedrockchannels (Fig. 3). This assumption is plausible and is adoptedsince there is a lack of direct relevant data. (ii) The sidewarddeflection length of bedload is largest at the edge of allu-vial patches or bars in the direction of the uncovered bedrock(Fig. 4). This assumption is made for three reasons. First, thebedrock is typically smoother than the alluviated section andprovides less impediment to particle movement, in particularto sideward deflection toward the uncovered part of the crosssection (cf. Chatanantavet and Parker, 2008; Ferguson et al.,2017; Hodge et al., 2011, 2016). Second, at the edge of bars,the alluvium provides roughness elements that can lead tosideward deflection (cf. Beer et al., 2017; Fuller et al., 2016).Third, at this point the velocity vector of the bedload par-ticles has a large cross-stream component; in fact, it is at itsmaximum (Fig. 3). In a channel with steady-state width, bed-load particles at this point should just fail to reach the wall,and we can assume that the sideward deflection length scaled is approximately equal to the uncovered width (Fig. 3). Atsteady state, we therefore expect that the following relationholds:
Cc =Wc
W=W − d
W= 1−
d
W. (2)
Using the equation for critical cover (Eq. 2), we can relatechannel width to vertical erosion rate using one of the estab-lished models for incision (e.g. Auel et al., 2017; Sklar andDietrich, 2004). I assume a sediment-flux-dependent erosionlaw, including tools and cover effect, of the form
E = kQs
W(1−C) . (3)
Here, E is the vertical erosion rate and k is a parameter thatdescribes the erodibility of the rock. As before, Qs is theupstream bedload sediment supply. Note that in the origi-nal saltation–abrasion model (Sklar and Dietrich, 2004), kdepends explicitly on hydraulics, but, consistently, in all ofthe field and laboratory studies where all relevant parame-ters have been measured, this dependency has not been found(Auel et al., 2017; Beer and Turowski, 2015; Chatanantavetand Parker, 2009; Inoue et al., 2014; Johnson and Whip-ple, 2010). At steady state, C=Cc. Substituting Eq. (2) intoEq. (3) and solving for width, we obtain an equation for the
Figure 3. Top: schematic drawing of the top view of the channelwith alternating gravel bars (dark grey), thalweg and main waterpathway (light grey), and main bedload path way (transparent darkgrey) after Bunte et al. (2006). Uncovered bedrock is depicted inwhite. Bottom: cross section across the centre of a bar (dashed blackline in the top view), where the bedload path crosses from the barinto the uncovered channel. This cross section is relevant for settingthe reach-scale channel width, since the sideward deflection of bed-load particles toward the left-hand wall should be maximized (cf.Fig. 4). At steady state, the uncovered width within the cross sec-tion should be equal to the sideward deflection length scale d, andthe relation d + Wc = W should hold (cf. Fig. 1; Eq. 2).
Figure 4. The potential sideward deflection distance is larger overalluvium than bedrock, since roughness elements facilitate sidewarddeflection of moving particles. However, the same roughness ele-ments block the path of the deflected particles, thus limiting thetotal distance. The largest deflection distances occur at the bound-ary between alluvium and bedrock towards the bedrock bed. Onlywhere the particle stream intersects this point can large sidewarddeflection distances be achieved.
steady-state width of bedrock channels.
W =
√kQsd
E(4)
2.2 Channel bed slope
To extend the argument to channel bed slope, an additionalequation is needed relating bed cover to sediment supply andtransport capacity. Several equations have been suggested inthe literature, including the linear decline model (Sklar andDietrich, 2004) and the negative exponential (Turowski etal., 2007). Recently, Turowski and Hodge (2017) derived a
J. M. Turowski: Alluvial cover controlling of bedrock channels 35
model of the form
C =
(1− e−
QsM0UW
)Qs
Qt. (5)
Here, e is the base of the natural logarithm, U is the aver-age bedload particle speed and M0 is the minimum mass perbed area necessary to completely cover the bed, which is de-pendent on grain size (Turowski, 2009; Turowski and Hodge,2017). Note that Eq. (5) reduces to the linear decline modelat high sediment supply, i.e. for large Qs.
We can write the bedload transport capacity per unit widthas a power function of both discharge Q and channel bedslope S (e.g. Rickenmann, 2001; Smith and Bretherton,1972):
Qt
W=KblQ
mSn. (6)
Here, Kbl is a constant, and it has been argued that the expo-nents m and n typically take values between 1 and 4 (Barryet al., 2004; Smith, 1974). Note that in Eq. (6), the thresholdof the motion of bedload has been neglected. Such a thresh-old is generally accepted to be relevant for bedload motion(e.g. Buffington and Montgomery, 1997) and will becomeimportant when linking sinuosity to storm frequency. Assum-ing steady state at the critical cover Cc, substituting Eqs. (2)and (6) into Eq. (5), and solving for S, we get
S =
(1− e−
QsM0UW
)1/n(Qs
Kbl (W − d)
)1/n
Q−mn . (7)
2.3 Sinuosity
At a given location, lateral erosion and therefore the devel-opment of curvature and sinuosity is of course dependent onlocal conditions such as the channel width, bed slope andlong-stream curvature (e.g. Cook et al., 2014; Howard andKnutson, 1984; Inoue et al., 2016). But rather than trying topredict the detailed evolution of the planform pattern, here Ipropose a reach-scale view of sinuosity development. As isconventional, sinuosity σ is defined as the ratio of the totalchannel length LC to the straight length LV from end to end.Note that this is equivalent to the ratio of valley slope SV tochannel bed slope S.
σ =LC
LV=SV
S(8)
Sinuosity can only increase if the walls of the channel areeroded. Thus, the rate of change of sinuosity dσ/dt should bezero when dW/dt is negative. Sinuosity development com-mences at the same critical cover Cc that marks the transi-tion from channel narrowing to widening, and dσ/dt shouldbe positive when dW/dt is also positive (Fig. 2). However,we need to slightly adjust the picture that has been advancedin Sect. 2.1, since instead of a straight channel, we are now
Figure 5. Schematic illustration of the thalweg (light grey) andgravel bedload path (dark grey) through a meandering channel, afterthe observations of Dietrich and Smith (1984) and Julien and An-thony (2002). Uncovered bedrock is depicted in white and gravelpoint bars in dark grey. Flow is from left to right. Dotted lines showthe relevant cross section for particle deflection. Areas that are pre-sumably affected by bedload particle deflection and should lead towall erosion are shaded in light grey. The dashed line is placed atthe inflection point of the channel centre line.
dealing with a curved channel. Further, channel curvature isvarying along the stream. As before, lateral erosion shouldstop once the channel walls are outside the reach of particleimpacts. Due to curvature, particle trajectories detach fromwater flow lines and wall erosion rates can be expected to behighest in regions with the highest curvature (e.g. Cook etal., 2014; Howard and Knutson, 1984). Point bars develop inthe inside bends, providing roughness for sideward deflection(Fig. 5). Substantial particle impacts can thus be expected inthe outside bends, probably a little downstream of the bendapex (cf. Fig. 5). The rest of the argument can stay essen-tially the same: lateral erosion stops once the bedrock wall isjust outside the reach of the deflected particles. The bedrockchannel is driven to a steady state at which C=Cc. At thispoint, sinuosity development ceases and the channel essen-tially stalls itself in its active meandering. Treating valleyslope as an independent parameter, Eq. (8) can be substitutedinto Eq. (7) and solved for sinuosity to obtain
σ =
(1− e−
QsM0UW
)1/n(Kbl (W − d)
Qs
)1/n
SVQmn . (9)
3 Comparison to observations
In this section, I will compare the model to field and labo-ratory observations. First, I will interpret the experiments ofShepherd (1972) in light of the arguments that lead to themodel equations. Then, I will compare field observations tothe predictions by the equations. Since for most field sitesmany essential parameters are not known, I will focus on ac-cepted scaling relations. Lague (2014) has summarized theavailable data for geometry and dynamics of bedrock chan-nels and has identified six lines of evidence that any model
36 J. M. Turowski: Alluvial cover controlling of bedrock channels
needs to match. Two of these are related to transient channeldynamics and knick point migration. Since the model devel-oped in the present paper is only concerned with steady-statechannels, the remaining lines of evidence, namely slope–areascaling, slope–erosion rate scaling, width–area scaling, andwidth–erosion rate scaling, are discussed below. To these Iadd the two scaling relations for the sinuosity of channels –sinuosity–erodibility scaling and sinuosity–storm-frequencyscaling – as observed by Stark et al. (2010).
For the comparison with field data, I use six data sets thatinclude information on erosion rates, with scaling relation-ships as compiled by Lague (2014) (Table 1). Two of thesedata sets arise from studies of rivers crossing a fault: theBakeya, Nepal (Lavé and Avouac, 2001), and the Peikangriver, Taiwan (Yanites et al., 2010). The data for the Bagmati,Nepal (Lavé and Avouac, 2001), were not used, since a trib-utary joins the stream within the studied reached, supplyingunknown amounts of both water and sediment and therebyaltering boundary conditions (see Lague, 2014; Lavé andAvouac, 2001; Turowski et al., 2009). Four of the data setsarise from studies comparing different catchments that arethought to be in a topographic steady state along a gradient inuplift rate with otherwise comparable conditions. These arechannels from the Siwalik Hills, Nepal (Kirby and Whipple,2001; reanalysed by Wobus et al., 2006b), the MendocinoTriple Junction (Snyder et al., 2000), Eastern Tibet (Ouimetet al., 2009) and the San Gabriel Mountains (DiBiase et al.,2010). I did not use the data from the Santa Inez Mountains(Duvall et al., 2004), since a lack of coarse bedload in thesemudstone channels has been reported (Whipple et al., 2013).There, impact erosion may not be the dominant erosion pro-cess, which could alter channel processes, morphology anddynamics. The channels studied by Tomkin et al. (2003)and Whittaker et al. (2007), draining catchments with stronglong-stream gradients in uplift rate, are under-constrained forthe purpose of model comparison, since the variation in ero-sion rates and therefore sediment supply along the stream isunknown.
For parts of the discussion it is useful to work with two ap-proximations for the cover equation, Eq. (5), both for the sakeof algebraic simplicity and ease of argument. First, in thetools-dominated domain, cover is scarce and bedrock erosionrate is controlled by the availability of tools. Then, Qs/W issmall and the exponential term can be approximated with afirst-order Taylor expansion, reducing Eq. (5) to
Ctools =Q2
sM0UWQt
. (10)
Then, we can reduce the first term in the slope and the sinu-osity equations, Eqs. (8) and (10), which yields(
1− e−Qs
M0UW
)≈
Qs
M0UW. (11)
Second, in the cover-dominated domain, tools are abun-dant, but most of the bed is covered. Then, the erosion rate is
set by the fraction of the exposed bedrock. Sediment supplyper unit widthQs/W is large, the exponential term vanishes,and we retrieve the linear model (Sklar and Dietrich, 2004).
Ccover =Qs
Qt. (12)
Then, the first term in the slope and the sinuosity equations,Eqs. (8) and (10), reduces to 1.(
1− e−Qs
M0UW
)≈ 1 (13)
The cover-dominated approximation (Eqs. 12 and 13) islikely most relevant for the data discussed here. It is knownthat many actively incising bedrock rivers exhibit substan-tial cover at least at low flow (Meshkova et al., 2012; Tinklerand Wohl, 1998; Turowski et al., 2008b, 2013), and it seemslikely that for many rivers the sideward deflection lengthscale d is much smaller than the channel width (formally,W � d, leading to W − d ≈ W ), implying substantial coverat steady state. Therefore, it can be expected that the tools-dominated approximation (Eqs. 10 and 11) is only relevantfor small headwater streams or for channels that do not re-ceive much coarse sediment, for example due to an upstreamreservoir.
3.1 Shepherd’s (1972) experiment
Shepherd’s observations have been described in detail in theintroduction. From a model perspective, consider a streamthat re-incises its bed after a base level drop. At constantsediment supply, as the stream incises, bed slope and there-fore transport capacity decreases. As a result, cover increases(Eq. 5). At some point the critical cover Cc is exceeded andthe stream starts active meandering. Meandering lengthensthe flow path and therefore also decreases bed slope andtransport capacity. The subsequent increase in cover leads, atsome point, to full cover stopping vertical incision. Once thesteady-state width is reached, lateral erosion drops to zero.Then, the stream also stops active meandering. It essentiallystalls itself and reaches a steady state for sinuosity. The de-scribed scenario is equivalent to the one observed by Shep-herd (1972), although the stalling phase was not reached inhis experiments.
3.2 Channel width
A number of studies report the sensitivity of channel widthto uplift rate (for summaries of the available data, see Lague,2014; Turowski et al., 2009; Whipple, 2004; Yanites andTucker, 2010). Several different behaviours have been ob-served (see also Table 1). In comparisons of channels incatchments that differ only by uplift rate, channel width wascomparable at similar drainage areas, indicating that therewas no response to uplift rate (Snyder et al., 2003; DiBiase
Channels crossing Observations Bakeya, Nepal −0.63 0.49 Lavé and Avouac (2001)a fault Peikang, Taiwan −0.42 0 Yanites et al. (2010)
Model prediction Tools-dominated −0.5 0.12–0.47Cover-dominated −0.5 0.07–0.33
Steady-state Observations Eastern Tibet Not measured 0.65 Ouimet et al. (2009)catchments San Gabriel Mountains 0 0.49 DiBiase et al. (2010)
Mendocino Triple Junction 0 0.25 Snyder et al. (2003)Siwalik hills Not measured 0.93 Kirby and Whipple (2001);
Wobus et al. (2006b)Model prediction Tools-dominated 0 0.27–1.05
Cover-dominated 0 0.14–0.67
and Whipple, 2011). In another study, Duvall et al. (2004)found narrower channels in catchments with higher upliftrates, but this could also be related to the lack of coarse bed-load in the mudstone channels (Whipple et al., 2013). Sim-ilarly, some channels display a typical width–area scalingdespite strong gradients in uplift rate (Tomkin et al., 2003;Whittaker et al., 2007). In contrast, channels crossing an up-lifting fault block tend to narrow (Lavé and Avouac, 2001;Yanites et al., 2010).
According to the proposed model, steady-state channelwidth scales with the square root of the product of bedloadsupply Qs, erodibility k and sideward deflection length scaled and inversely with the square root of the vertical incisionrate E (Eq. 4). The different response of channel width instudies comparing different channels in areas with gradientsin uplift rate (no channel narrowing) and those that lookedat single channels crossing an uplifting fault block (channelnarrowing) can be explained by the role of sediment flux. Iwill discuss the latter case first.
When a channel crosses from a region without uplift intoan uplifting fault block, water discharge and sediment loadstay the same, provided there are no tributaries or major hill-slopes sediment sources. Thus, in the width equation (Eq. 4),bedload supply Qs is constant and the channel responds byincreasing erosion rate E to match the increased uplift rate.Provided that k and d are independent of erosion rate, thechannel narrows and channel width should scale with in-cision rate to the power of −1/2. Two of the cases men-tioned above allow a direct evaluation of this prediction.In the Bakeya River (Lavé and Avouac, 2001), the width–erosion rate scaling exponent is −0.63, and in the PeikangRiver (Yanites et al., 2010), the scaling exponent is −0.42(Table 1); both are close to the predicted value of −1/2.
In catchments in a topographic steady state, the chan-nel geometry adjusts such that the long-term incision ratematches the long-term uplift rate or base level lowering rate.
Averaged over the catchment, bedload supply can be writtenin terms of erosion rate E and catchment area A.
Qs = βEA (14)
Here, β is the fraction of material that contributes to bedrockerosion, i.e. the bedload fraction. The steady-state channelwidth Eq. (4) then becomes
W =√kβdA. (15)
As vertical incision rate E cancels out, steady-state channelwidth in this case is independent of uplift rate. This is inagreement with field observations (Table 1). Equation (15)also predicts the typical scaling of channel width W withthe square root of drainage area A. However, it is likely thatboth the gravel bedload fraction β and the sideward deflec-tion length scale d vary in a systematic fashion with drainagearea. The bedload fraction tends to decrease with increasingdrainage area (e.g. Turowski et al., 2010), possibly even tothe extent that bedload supply Qs is independent of drainagearea (see Dingle et al., 2017). There are additional complica-tions that arise from non-linear averaging of sediment supplyboth with varying floods and stochastically varying bedloadsupply. Further, the bedload fraction β is likely dependenton erosion rate E, in a currently unknown way. At the mo-ment little is known about how d varies along a stream. I willreturn to this point in the discussion.
3.3 Channel bed slope
A power-law scaling of slope with drainage area with an ex-ponent of−1/2 is widely assumed to be indicative of steady-state bedrock channels.
S = ksA−θ (16)
This relationship is known as Flint’s law (Flint, 1974), al-though it was studied by Hack (1957) at an earlier point. The
38 J. M. Turowski: Alluvial cover controlling of bedrock channels
pre-factor ks is called the steepness index, and the exponentθ is called the concavity index. For the concavity index, arange of values of 0.4–0.6 is often reported (Lague, 2014).Whipple (2004) gives a range of 0.4–0.7 for actively incis-ing bedrock channels in homogenous substrates with uni-form uplift, while higher concavities (0.7–1.0) are associatedwith decreasing uplift rates in the downstream direction. Us-ing data from catchments where erosion rate have been con-strained using cosmogenic nuclides, Harel et al. (2016) founda median value of the concavity index of 0.52± 0.14, with asimilar range as reported by Whipple (2004). It seems, there-fore, that the observed variability in the value of the concav-ity index is higher than is generally acknowledged, with ob-served values as low as 0.4 and as high as 1. In comparisonsof channels in steady-state landscapes, the steepness indexks has been observed to increase with incision rate accordingto a power law, with an exponent ranging from 0.25 to 0.93(Table 1), derived from four data sets (Lague, 2014). The twochannels crossing a fault block exhibit different scaling. TheBakeya (Lavé and Avouac, 2001) shows a positive relation-ship with an exponent of about 0.49, while for the Peikang(Yanites et al., 2010), little to no slope changes in responseto uplift have been reported.
The brief summary of observations above implies that amodel should be able to account for the following observa-tions. (i) Slope should decrease with drainage area accord-ing to a power law with an exponent value varying betweenabout 0.4 and 0.7. (ii) The exponent may be altered if thereare gradients in uplift rate along the stream; in particular, adownstream decrease in uplift may drive the concavity indexup to higher values of up to about 1. (iii) In channels drain-ing catchments in a topographic steady state, the steepnessindex should increase with uplift rate according to a powerlaw with an exponent value varying between about 0.25 and1.0. (iv) In channels crossing a fault block, slope may or maynot increase in response to uplift.
Often, the concavity index in the slope–area relationshipis related to a slope–discharge scaling by assuming that dis-charge scales with drainage area following a relationship ofthe form
Q= khAc. (17)
Here, kh and c are catchment-specific values describing thehydrology. In particular, the exponent c takes a value of 1 ifthe exchange of water with groundwater storage and evapo-transpiration are spatially uniform in the catchment (e.g. Sny-der et al., 2003). For natural data, the value of c is dependenton the discharge chosen for the regression. For the long-termmean annual discharge, various effects should average outand c should be close to 1 (Dunne and Leopold, 1978, ascited by Snyder et al., 2003). Leopold et al. (1964) reportedvalues between 0.70 and 0.75 for bankfull discharge. Whentransforming the observed values of the concavity index ofthe slope–area scaling to an exponent of the slope–discharge
relationship, we thus obtain a range of values for the slope–discharge exponent of 0.4–1.0 for steady-state channels inuniform conditions and 0.7–1.4 for channels with a down-stream decrease in uplift rate.
In the model equation (Eq. 7), slope scales with dischargeto a power of −m/n. Many bedload transport equations canbe written in the form of Eq. (6) (Smith and Bretherton,1972), and the theoretical values of m and n depend on thechosen equation. For example, the Einstein (1950) bedloadequation yields m = n = 2 (Smith and Bretherton, 1972),while Meyer-Peter and Müller (1948) type equations yieldm= 1 and n = 1.5 (Rickenmann, 2001). However, in thelatter case, the linear scaling arises only if the threshold ofbedload motion is neglected and is thus valid only for largefloods. Rickenmann (2001) argued that n = 2 gives a betterfit for both laboratory and field data at stream gradients largerthan 3 %. However, he also included relative roughness as aseparate predictor, which is implicitly dependent on slope.If written out explicitly, the dependence on slope should bestronger, with values of n potentially much larger than 2 (seealso Nitsche et al., 2011; Schneider et al., 2015). Measuredm values are usually much larger than those derived frommodels. For example, Bunte et al. (2008) reported m valuesranging from about 7.5 to 16, using data obtained during thesnowmelt period of American streams with portable bedloadtraps. Analysing bedload data sampled with Helley–Smithpressure difference samplers from a large number of streams,Barry et al. (2004) found values of m in the range of about1.5–4.0. They used drainage area instead of slope in theirtransport equation, and the data given in their paper do not al-low a re-evaluation in terms of slope. Nevertheless, a regres-sion of channel bed slope of the sites against drainage areayields an exponent of −0.48, giving an estimate of n≈ 7.1.From the mentioned cases, it is clear that depending on thechoice of equation or data set, a wide range of values forthe m and n scaling exponents can be obtained. Finally, itneeds to be noted that most bedload data and bedload trans-port equations in the literature have been derived for channelswith a mobile bed. Bedload equations specifically for natu-ral bedrock channels are not known to the author. In additionto the explicit relationship of slope and discharge, slope isimplicitly related to discharge via sediment supply, channelwidth and the sideward deflection length scale, all of whichcould depend on discharge or drainage area.
Of the discussed approaches, the field data evaluation byRickenmann (2001) may be most appropriate for the purposeat hand, since the data were derived from long-term mon-itoring of deposition in retention basins. The timescale ofthe data is thus closer to the timescales of bedrock erosionand channel adjustment than the near-instantaneous measure-ments used for example by Barry et al. (2004). This wouldyield values ofm= 1 and n= 2 and a ratiom/n= 0.5 (Rick-enmann, 2001). For the remainder of the discussion, I willuse this case as standard, as well as a range of n values of
J. M. Turowski: Alluvial cover controlling of bedrock channels 39
1.5–7 for evaluating possible ranges of the values of scalingexponents.
The equations and the discussion are considerably sim-plified in the tools- or cover-dominated approximations (seeEqs. 10–13). In the tools-dominated case, channel bed slopeis given by
Stools =
(Q2
sM0UKblW (W − d)
)1/n
Q−mn . (18)
Here, we can recognize two different cases. First, considernarrow headwater channels. There, the sideward deflectionlength scale d is of the order of the channel width W . As aresult, slope depends strongly on the actual values of d andW and their scaling with other morphological parameters,e.g. bed roughness. I will not consider this case further, asthere are few relevant data available. Second, consider a widechannel carrying little coarse sediment, for instance due to anupstream reservoir. Then, W � d and Eq. (18) reduces to
Stools =
(Q2
s
M0UKblW 2
)1/n
Q−mn . (19)
Since bedload particle speed U is dependent on hy-draulics, there is an implicit dependence of U on slope anddischarge, which needs to be taken into account. With stan-dard assumptions on flow velocity and shear stress (Ap-pendix A), Eq. (19) becomes
Stools = ktools
(E
kd
) 3+α4n+α+1
(Qs)5−α
4n+α+1 (Q)−4m−2α+24n+α+1 . (20)
Here, ktools is assumed to be constant (see Eq. A9, Ap-pendix A), and α is a constant that typically takes a valueof 0.6 (e.g. Nitsche et al., 2012). In the case of a steady-statechannel crossing an uplifting fault block, Qs and Q can beconsidered constant and only E varies. In this case, the dis-charge exponent is equal to −0.5 as long as m/n = 1/2. Forn= 1.5, the dependence on erosion rate and erodibility yieldsan exponent of 0.47, with decreasing values as n increases (itreaches 0.375 for n= 2, 0.20 for n= 4 and 0.12 for n= 7).For a channel in a steady-state landscape, we can substituteEq. (14) to obtain
Stools = ktools(βA)5−α
4n+α+1E8
4n+α+1 (kd)−3+α
4n+α+1
(Q)−4m−2α+24n+α+1 . (21)
Now, the exponent on erosion rate varies between 0.27 and1.05. As before, slope–area scaling cannot be evaluated in ameaningful manner, since the dependence of β and d on areais unknown.
In the cover-dominated case, Eq. (7) reduces to
Scover =
(Qs
KblW
)1/n
Q−mn =
(EQs
K2blkd
)1/2n
Q−mn . (22)
Here, I also used the approximation W � d , and channelwidth was eliminated using Eq. (4). For rivers crossing anuplifting fault block, where all parameters apart from erosionrate can be treated as constant, slope scales with the incisionrate E1/2n, with the exponent lying in the range of 0.07–0.33and using a range of n values of 1.5–7, as discussed above.For catchments in a topographic steady state, Qs can be ex-pected to scale linearly with erosion rate (Eq. 14), yielding aslope equation of the form
Scover =
(βAE2
K2blkd
)1/2n
Q−mn . (23)
In this case, the exponent on erosion rate yields the rangeof values of 0.14–0.67. The dependence on Qs introducesan additional dependence on area, affecting the slope–areaexponent. Assuming that Q is proportional to drainage area(c= 1) and m= 1 and n= 2, the slope–area exponent evalu-ates to −0.25. However, both bedload fraction and sidewarddeflection distance can be expected to scale with drainagearea in an unknown way, which would alter the relationship.In addition, if E varies systematically along the stream, theslope–area scaling will be affected. For example, if E de-creases in the downstream direction, it also decreases withincreasing drainage area, resulting in an increase in the con-cavity index. This is in line with observations.
In summary, the values for the scaling exponents for therelationship between slope and erosion rates for the differ-ent cases that have been discussed encompass the range ofobserved values (Table 1). All four observations regardingchannel bed slope, as outlined in the beginning of this chap-ter, can be obtained.
3.4 Sinuosity
Recapitulating the results of Stark et al. (2010), we expectsinuosity to increase both with increasing erodibility k andincreasing storm strike frequency. After substituting Eq. (4)into Eq. (9) to eliminate channel width and employing theapproximation W � d, the tools-dominated case gives
σtools =
(KblM0Ukd
QsE
)1/n
SVQmn . (24)
As before, the bedload particle speed U is dependent onslope and discharge. Accounting for this gives
σtools =SV
ktools
(kd
E
) 3+α4n+α+1
(Qs)α−5
4n+α+1
(Q−Qc)4m−2α+24n+α+1 . (25)
Here, I have also replaced discharge Q with effective dis-charge Q−Qc, subtracting a critical discharge for the on-set of bedload motion Qc (e.g. Buffington and Montgomery,
40 J. M. Turowski: Alluvial cover controlling of bedrock channels
1997; Rickenmann, 2001), which is important when consid-ering discharge variability (e.g. Lague et al., 2005; Molnar,2001) and thus sinuosity dependence on storm frequency. Inthe cover-dominated case, we get
σcover =
(K2
blkd
EQs
)1/2n
SV(Q−Qc)mn . (26)
For the following discussion, SV is treated as a constantbut could in principle be a function of local tectonics and,therefore, implicitly erosion rate. The expected scaling witherodibility is directly obvious from both Eqs. (25) and (26);sinuosity scales with k(3+α)/(4n+α+1) in the tools-dominatedcase and with k1/2n in the cover-dominated case. Since thereis currently no accepted way of measuring k, no quantitativedata exist and the comparison cannot go further.
Next, we link sinuosity to the variability in precipita-tion. The variability in forcing parameters is important forthreshold processes (e.g. Lague, 2010), and the only relevantthreshold process that we have considered is bedload trans-port. When considering variable forcing, mean dischargeneeds to be replaced by the effective discharge Qeff that de-termines bedload transport and incision on long timescales(e.g. Lague et al., 2005; Molnar, 2001). In general, if thethreshold discharge is higher than the mean discharge, ahigher discharge variability results in a higher effective dis-charge (Deal, 2017). In storm-driven catchments, such asthe streams on the Pacific Arc islands studied by Stark etal. (2010), geomorphically active floods are generally rare(e.g. Molnar, 2001) and erosion is limited to a few daysper year and often less, making this assumption valid. Vari-ability in discharge VQ scales with the frequency of largestorms FStorm (cf. Deal, 2017; Rossi et al., 2016). We thusfind a scaling that agrees with the observations of Stark etal. (2010):
σ ∼Qeff ∼ VQ ∼ FStorm. (27)
4 Discussion
4.1 Comparison to previous models
The model proposed here connects channel width, bed slopeand sinuosity to discharge, erosion rate and substrate erodi-bility, via the core variable of bed cover. It fills a gap withinthe available published models, as it is a 1-D reach-scalemodel constructed from considerations of the physics of bed-load transport and fluvial erosion, without the need of arbi-trary closing assumptions. I have used a fluvial bedrock ero-sion model (Eq. 3) that includes both tools and cover effectsand that is consistent with current process understanding (e.g.Beer et al., 2017; Fuller et al., 2016; Johnson and Whipple,2010; Sklar and Dietrich, 2004), as well as quantitative fieldand laboratory measurements (Auel et al., 2017; Beer andTurowski, 2015; Chatanantavet and Parker, 2009; Inoue et
al., 2014; Johnson and Whipple, 2010). The model presentedhere thus improves upon existing 1-D reach-scale modelsboth in the plausibility of the underlying assumptions and, ashas been shown in Sect. 3, in the predictive power concern-ing the observed scaling relationships. In addition, the modelis complete in the sense that it does not feature a lumped cali-bration parameter with obscure physical meaning. All modelparameters have a direct physical interpretation and can, atleast in principle, be measured in the laboratory or the field.
4.2 Sideward deflection of bedload
To further validate or refine the model, we need informationon some of the unconstrained parameters. In particular, weare missing observations on bedload paths in partially allu-viated beds and on sideward deflection of bedload particles.While no data are available on the former, at least some ini-tial observations have been reported on the latter. From lab-oratory observations, Fuller et al. (2016) argued that rough-ness dominantly controls sideward deflection of bedload inbedrock channels and therefore lateral erosion. This inter-pretation is supported by field data of Beer et al. (2017).For a full quantification of the model, the sideward deflec-tion length scale d would need to be measured for a realisticrange of boundary conditions, varying hydraulics, bed rough-ness, particle size and characteristics. To upscale the modelto the reach scale, we would need scaling relationships of bedroughness with drainage area or other morphological param-eters that vary along a stream. A comprehensive investigationof the controls on the scaling of bed roughness of bedrockchannels is not known to the author. An additional complica-tion arises from the role of alluvium. An alluviated bed is typ-ically rougher than bedrock (e.g. Chatanantavet and Parker,2008; Ferguson et al., 2017; Hodge et al., 2011, 2016), andthe effect of stationary sediment on a bedrock bed on side-ward deflection of moving particles has not yet been investi-gated.
We can obtain some tentative constraints on these scal-ing relationships by considering catchments in a topographicsteady state. I assume that, in the cover-dominated domain,sideward deflection length scale d and bedload fraction β aredependent on drainage area A according to a power law, withexponents a and b, respectively. The slope–area scaling canbe written as
Scover ∼
(β
d
)1/2n
A1
2n−cmn ∼ A
b−a2n A
12n−c
mn
= Ab−a+1
2n −cmn . (28)
Here, I used the hydraulic scaling (Eq. 17) to replace dis-charge with area. If we assume that the concavity index,which includes both the explicit and implicit dependence ondrainage area in Eq. (28), is equal to 1/2 and usem= 1, n= 2and c= 1 (see Sect. 3.2), we obtain b− a= 1. Similarly, as-suming that the width–area scaling in Eq. (15) should have
J. M. Turowski: Alluvial cover controlling of bedrock channels 41
an exponent of 1/2, from the width Eq. (15), we obtain
W ∼ (βd)1/2A1/2∼ A
a+b2 A1/2
= Aa+b+1
2 . (29)
This yields a+ b= 0. Solving, we obtain a = 1/2 andb=−1/2. This means that the sideward deflection length dincreases when moving downstream while the bedload frac-tion β decreases, both with the square root of drainage area.At least for the bedload fraction, this seems to be a plausiblevalue (see Turowski et al., 2010). For d , at first glance, an in-crease with drainage area seems somewhat surprising, sinceit is often assumed that roughness decreases in the down-stream direction (e.g. Ferguson, 2007; Nitsche et al., 2012).However, this assumption is made for alluvial channels andis related to downstream fining that is observed in many al-luvial streams (e.g. Parker, 1991). In a bedrock channel, itseems plausible that a progressive increase in cover leads toan overall increase in roughness when moving downstream.
4.3 Implications for stream-profile inversion
The theoretical framework of the stream power model hasbeen frequently used to obtain information about tectonic up-lift or fluvial erosion rates by stream-profile inversion (e.g.Kirby and Whipple, 2001; Wobus et al., 2006b). Within thestream power framework, the steady-state profile of bedrockchannels is given by
S =
(E
ke
)1/n′
A−cm
′
n′ . (30)
Here, ke is a lumped calibration parameter that is commonlyinterpreted to reflect bedrock erodibility. For the analysis, itis usually assumed thatm′= 0.5, n′= 1 and c= 1 (see Lague,2014), to obtain a concavity index equal to 1/2, althoughevidence points to n′ typically being larger than 1 (DiBiaseand Whipple, 2011; Harel et al., 2016; Lague, 2014). Then,slope is fitted with a power law against area and a value forE/ke can be derived. More sophisticated inversions exploitthe transient dynamics of models that can resolve erosionhistories and find separate fit solutions for both E and ke(e.g. Roberts and White, 2010). Comparing Eq. (30) to thefour slope equations obtained by the model (Eqs. 20–23), thesteady-state equations show the same power-law dependenceof slope S on drainage area A and erosion rate E, although,depending on the domain (cover- vs. tools-dominated) andthe type of forcing (crossing a fault or topographic steadystate), the scaling exponents differ. In particular, the relation-ship between the scaling exponent of slope with erosion rateand the scaling exponent on drainage area (the concavity in-dex) may be different to the one inferred from Eq. (30). Forexample, for steady-state catchments in the cover-dominateddomain, the scaling exponent on erosion rate evaluates to 1/n(Eq. 23), while the concavity index evaluates to (1+ cm)/n(Eq. 28; using b – a= 1). Further, the physical interpreta-tion of m and n is different from the interpretation of m′ and
n′. While in the stream power model, m′ and n′ are directlyrelated to the mechanics of fluvial bedrock erosion, in themodel proposed here, m and n are related to the mechanicsof bedload transport. Clearly, a wrong choice for the valueof n′, in particular, leads to incorrect estimates of erosionrates. If m′ and n′ are determined by bedload transport, assuggested here, n′ may fall in the plausible range between1.5 and 7 (see Sect. 3.2) and could be very different fromn′= 1 that is typically used when deriving tectonic informa-tion from stream-profile inversion.
4.4 The role of cover for sinuous bedrock channels
Here, I have argued that in streams where impact erosion isthe dominant fluvial erosion process, cover is the central vari-able that needs to be considered. Nevertheless, it can be ex-pected that bed cover modulates sinuosity development alsoin streams where other erosion processes are dominant. Ashas been argued by Johnson and Finnegan (2015), the dom-inant erosion process – slaking or impact erosion – deter-mines whether a particular stream actively meanders or notin their study region. However, even weak rock that can beworn away by clear water flow will not erode if it is coveredby a thick layer of sediment. Moreover, arguably, wetting–drying cycles are both less frequent and less efficient whenwater needs to flow through the pores of a gravel or sandlayer. Although the erosion mechanism may likely make cer-tain channels more prone to active meandering than others, Isuggest here that bed cover plays a role in all of them.
5 Conclusions
Based on the idea that relative sediment supply controlsbedrock channel meandering (Moore, 1926; Shepherd, 1972)and by making links to lateral erosion and channel widthevolution, a physics-based 1-D model of bedrock channelmorphology was constructed. The model correctly predictsthe observed scaling relations between channel width andslope, on the one hand, and discharge and erosion rate, on theother, as well as those between sinuosity, on the one hand,and erodibility and storm strike frequency, on the other. Inaddition, it yields plausible ranges of values of the expo-nent values and can explain why a channel should developto a steady-state sinuosity. The model is rooted in processphysics, is fully parameterized and does not include lumpedcalibration parameters. It therefore describes bedrock chan-nel morphology more completely than previously proposedmodels.
By predicting steady-state long profiles of bedrock chan-nels similar to the stream power model, the model proposedhere explains the success of the stream power model in de-scribing steady-state channel bed slope and its failure to ac-count for the scaling of width. In addition, it reconnects chan-nel long-profile analysis with the insights that have been ob-tained on the physics of fluvial bedrock erosion over the last
42 J. M. Turowski: Alluvial cover controlling of bedrock channels
2 decades. If the physical argument proposed here is correct,methods of stream-profile inversion to obtain data on erosionrate or tectonic history using the stream power model arebased on incorrect assumptions. The results obtained withthese methods are likely incorrect, especially if they wereused to derive uplift histories. A further interesting point isthat, if the model is correct, the scaling of width with ero-sion rate (Eq. 4) seems less complicated than that of slope(Eq. 7), since its dependence on the details of the hydraulicsand hydrology is less pronounced. This may indicate that tec-tonic information can be more robustly obtained from chan-nel width than from slope.
The model proposed here opens a new view to reach-scalebedrock channel morphology. Although the assumptions thathave been made are physically plausible, many of them are asyet untested and few data are available to constrain the valuesof and the controls on some of the key parameters, such asthe sideward deflection length scale. Nevertheless, the strongrooting of the model in process physics and its success incorrectly predicting scaling relationships of slope, width andsinuosity is encouraging and warrants further investigation.For a comprehensive evaluation of the model and the un-derlying assumptions, we need detailed investigations of thesediment dynamics in partially alluviated bedrock channels.In particular, this includes bedload transport equations forparticles moving over a bare bedrock bed, maps of bedloadparticle concentrations in the channel for various bed mor-phologies and flow conditions, and research into the controlson sideward deflection of moving particles.
Data availability. No data sets were used in this article.
J. M. Turowski: Alluvial cover controlling of bedrock channels 43
Appendix A
In the tools-dominated domain, the channel bed slope isgiven by Eq. (19):
Stools =
(Q2
s
M0UKblW 2
)1/n
Q−mn . (A1)
Here, the bedload particle speed U depends on shear stressand therefore slope and discharge. Based on laboratory flumemeasurements, Auel et al. (2017) gave an equation (theirEq. 19) for particle speed as a function of shear stress τ , in-cluding various previous measurements, both over bedrockand alluvial beds.
U = 1.46(
1ρ
(τ
τc− 1
))1/2
(A2)
Here, tauc is the critical shear stress for the onset of bed-load motion. To eliminate shear stress, I use the DuBoysequation and the water continuity equation:
τ = ρgHS, (A3)Q=WHV. (A4)
Here, H is the water depth. Water flow velocity V can becomputed by the variable power flow resistance equation,which can be expressed as a function of slope, discharge andwidth (Ferguson, 2007; Nitsche et al., 2012):
V = kV(gS)1−α
2 R1−3α
2
(Q
W
)α. (A5)
Here, R is a measure of bed roughness with dimensions oflength, for example the standard deviation of the bed surface(e.g. Nitsche et al., 2012), and α≈ 0.6 is a constant. Shearstress can then be written as
τ =ρ
kV(gS)
α+12 R
3α−12
(Q
W
)1−α
. (A6)
For substitution into Eq. (A1), I neglect the threshold (i.e.τ/τc− 1≈ τ/τc) to obtain
44 J. M. Turowski: Alluvial cover controlling of bedrock channels
Appendix B: Notation
A Drainage area (m2).Acover Covered bed area (m2).Atot Total bed area (m2).a Scaling exponent, d −A.b Scaling exponent, β −A.C Fraction of covered bed.Cc Critical cover.c Scaling exponent, Q−A.d Sideward deflection length scale, reach (m).dx Sideward deflection length scale, cross section (m).dxy Sideward deflection length scale, at a point (m).E Vertical erosion rate (m s−1).e Base of the natural logarithm.FStorm Storm strike frequency (s−1).g Acceleration due to gravity (m s−2).H Water depth (m).Kbl Bedload transport efficiency (kg m−3m−1 sm−1).k Erodibility (m2 kg−1).ke Erodibility in stream power model (m1−3m sm−1).kh Hydrology coefficient (m3−2c s−1).ks Steepness index (m2θ ).ktools Lumped constant, tools-dominated channel slope.kV Velocity coefficient.L Reach length (m).LV Straight length from reach start to end (m).M0 Minimum mass per area necessary to cover the bed (kg m−2).m Discharge exponent in bedload equation.m′ Discharge exponent in the stream power model.n Slope exponent in bedload equation.n′ Slope exponent in the stream power model.Q Water discharge (m3 s−1).Qc Critical discharge for the onset of bedload motion (m3 s−1).Q∗c Relative sediment supply at the critical cover.Qeff Effective discharge (m3 s−1).Qs Upstream sediment mass supply (kg s−1).Q∗s Relative sediment supply; sediment transport rate over transport capacity.Qt Mass sediment transport capacity (kg s−1).R Bed roughness length scale (m).S Channel bed slope.Scover Channel bed slope predicted in the cover-dominated approximation.Stools Channel bed slope predicted in the tools-dominated approximation.SV Valley slope.U Bedload speed (m s−1).V Water flow velocity (m s−1).VQ Discharge variability parameter.W Channel width (m).Wcover Covered length within the channel width (m).α Scaling exponent, V −Q.β Fraction of sediment transported as bedload.θ Concavity index; scaling exponent S−A.ρ Density of water (kg m−3).σ Sinuosity.σcover Sinuosity predicted in the cover-dominated approximation.σtools Sinuosity predicted in the tools-dominated approximation.τ Bed shear stress (N m−2).τc Critical bed shear stress at the onset of bedload motion (N m−2).
J. M. Turowski: Alluvial cover controlling of bedrock channels 45
Competing interests. The author declares that he has no conflictof interest.
Acknowledgements. I thank Joel Scheingross and Eric Dealfor insightful discussions. The model presented here is rootedin discussions with Colin Stark and Jonathan Barbour about adecade ago. The author was prompted to take up the problem ofbedrock channel sinuosity again during a field visit to the ChineseTien Shan in 2016, studying rivers affected by active folding withKristen Cook. Mitch D’Arcy, Joel Scheingross and Aaron Bufeas well as two anonymous reviewers looked through a previousversion of this paper, and their helpful comments led to tremendousimprovements.
The article processing charges for this open-accesspublication were covered by a ResearchCentre of the Helmholtz Association.
Edited by: Daniel ParsonsReviewed by: two anonymous referees
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